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Adaptive Dynamics of Temperate Phages

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Lysis means the phage makes many copies of itself and releases the new phages by ... Lysis looks like this ... p) = probability of lysis. i = induction rate. ... – PowerPoint PPT presentation

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Title: Adaptive Dynamics of Temperate Phages


1
Adaptive Dynamics of Temperate Phages
2
Introduction
  • Phages are viruses which infect bacteria
  • A temperate phage can either replicate lytically
    or lysogenically
  • Lysis means the phage makes many copies of itself
    and releases the new phages by bursting the
    bacteria open. Bacteria is destroyed.
  • Lysogeny means the phage inserts its DNA into the
    bacterial DNA and is replicated passively when
    the bacteria divides. Bacteria (lysogen)
    survives.
  • Lysogens can later be induced, i.e. phage DNA
    extricates itself from the bacterial DNA and
    carries out lysis.

3
Lysis looks like this

4
The populations in the model
  • R resources
  • S sensitive bacteria
  • P1 phage strain
  • P2 another phage strain
  • L1 lysogens of phage P1
  • L2 lysogens of phage P2
  • The only differences between P1 and P2 are that
    they have different probabilities of lysogeny and
    different induction rates.

5
Some important parameters
  • ? chemostat flow rate
  • d adsorption rate
  • p probability of lysogeny
  • (1-p) probability of lysis
  • i induction rate
  • ß burst size

6
The Model

7
Invasion of resident strain by a mutant
  • Suppose P1 is the resident phage.
  • Assume that the system has reached its
    equilibrium (R, S, L1, P1)
  • Can P2 invade?

8
Linearization around the equilibrium
  • To see if P2 can invade, consider the linearized
    system
  • P2 can invade if there is a positive eigenvalue

9
The fitness function
  • It turns out that there will be at least one
    positive eigenvalue as long as the following
    condition is satisfied


10

Q, µ, and ?

11
Introducing a trade-off function
  • Now let i f(p)
  • Fitness function becomes

i
p
12
Evolutionary singularities
  • At an evolutionary singularity (p1p2p), the
    first order partial derivatives of the fitness
    function with respect to p1 and p2 will be equal
    to zero
  • Differentiating sp1(p2) with respect to p2 and
    setting equal to zero
  • So at a singularity p, we must have either

or
13
Identifying evolutionary singularities

14
Branching points

15
Evolutionary branching
  • Let p be an evolutionary singularity
  • Let
  • Then p will be a branching point if
  • (i) bgt0 (i.e. p is not ESS)
  • (ii) (a-b)gt0 (i.e. p is CS)

16
Differentiating with respect to p2
  • Let b be the second order derivative of the
    fitness function with respect to p2, evaluated at
    the singularity p
  • Then

17
Differentiating with respect to p2
  • Let a be the second order derivative of the
    fitness function with respect to p1, evaluated at
    the singularity p.
  • It turns out that

18
  • Suppose bgt0 (i.e. singularity is not ESS)
  • For evolutionary branching, we also need (a-b)gt0
    (i.e. singularity is CS).
  • From previous slide
  • So we need to find the derivative of µ at the
    singularity

19
Finding the derivative of µ
  • Start from the resident ODEs at equilibrium

20
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21
Derivative of µ is zero
  • Remember that µ(p)dS(p)P(p)/L(p)
  • We know the derivatives of S, P and L are all
    zero
  • So by the quotient rule, the derivative of µ must
    also be zero.
  • So from
  • we find that
  • i.e. branching is not possible.

22
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